Introduction to Braid Groups Joshua Lieber VIGRE REU 2011 University of Chicago ABSTRACT. This paper is an introduction to the braid groups in- tended to familiarize the reader with the basic definitions of these mathematical objects. First, the concepts of the Fundamental Group of a topological space, configuration space, and exact sequences are briefly defined, after which geometric braids are discussed, followed by the definition of the classical (Artin) braid group and a few key results concerning it. Finally, the definition of the braid group of an arbitrary manifold is given. Contents 1. Geometric and Topological Prerequisites 1 2. Geometric Braids 4 3. The Braid Group of a General Manifold 5 4. The Artin Braid Group 8 5. The Link Between the Geometric and Algebraic Pictures 10 6. Acknowledgements 12 7. Bibliography 12 1. Geometric and Topological Prerequisites Any picture of the braid groups necessarily follows only after one has discussed the necessary prerequisites from geometry and topology. We begin by discussing what is meant by Configuration space. Definition 1.1 Given any manifold M and any arbitrary set of m points in that manifold Qm, the configuration space Fm;nM is the space of ordered n-tuples fx1; :::; xng 2 M − Qm such that for all i; j 2 f1; :::; ng, with i 6= j, xi 6= xj. If M is simply connected, then the choice of the m points is irrelevant, 0 ∼ 0 as, for any two such sets Qm and Qm, one has M − Qm = M − Qm. Furthermore, Fm;0 is simply M − Qm for some set of M points Qm. In this paper, we will be primarily concerned with the space F0;nM. 1 2 Another space useful for our purposes is obtained by taking Bm;nM = Fm;nM=Sn, which just takes any fx1; :::; xng 2 Fm;nM and sets all permutations of those coordinates equal. Furthermore, there is a can- nonical projection p : Fm;nM ! Bm;nM. Definition 1.2 Given two continuous maps f; g : X ! Y , where X and Y are topological spaces, f and g are said to be homotopic if there exists a continuous function H : X × [0; 1] ! Y such that H(x; 0) = f(x) and H(x; 1) = g(x). This map H is referred to as a homotopy from f to g. A homotopy between maps is simply a method of continuously deform- ing one map into the other. For our purposes, it is most important in light of the following definitions. Definition 1.3 A path in a topological space X is a continuous map f : [0; 1] ! X. If f(0) = f(1), then f is referred to as a loop. A space is called path connected if any two points in the space are connected by a path. Definition 1.4 Given a based, path connected topological space (X; x), the fundamental group of that space, π1(X), is the set of all loops start- ing and ending at x up to homotopy such that the endpoints are fixed (i.e. h(0; t) = h(1; t) = x for all t 2 [0; 1]). There is a manner in which one may transform loops from one point into loops from the chosen basepoint. Given the space (X; x) and another point y, one may take a loop γ from y to y, and transform it into a loop from x to x. Simply take a path η from x to y, and analyze the composition ηγη−1. Clearly, this corresponds to a loop from x to x. Analogously, one may transform loops from x to x into loops from y to y. Hence, the choice of basepoint is immaterial and is omitted from the notation. Proposition 1.5 The fundamental group of a topological space X is, in fact, a group. 3 Proof. In order to show this, one needs to establish a multiplicative structure on π1(X). Take f; g 2 π1(X). Then one may define f · g as follows: 1 f(2t) : 0 ≤ t ≤ 2 (f · g)(t) = 1 g(2t − 1) : 2 ≤ t ≤ 1 The associativity of this product may be easily checked. In addition, there exists an identity element id(t) = x, and inverses, which may be −1 written as f (t) = f(1 − t) for any f 2 π1(X). One may easily verify that the group properties hold. This is one of the most important concepts in Algebraic Topology. It is particularly important to us for reasons which will be made apparent in later sections. Definition 1.6 An exact sequence is a chain of groups and homomor- phisms between them such that the image set of one homomorphism in the chain is equal to the kernel of the following one. In other words, it is a chain f1 f2 fn−1 G1 −! G2 −! · · · −! Gn such that for all k 2 f1; :::; n−1g, one has im(fk) = ker(fk+1). A short exact sequence is one of the form f g 1 −! A −! B −! C −! 1; where 1 represents the trivial group. This forces f to be a monomor- phism and g to be an epimorphism (in the nonabelian case, there is the additional requirement that f is the inclusion of a normal subgroup). Furthermore, this in turn induces an isomorphism C ∼= B=A. Now, there are two important lemmas about exact sequences which must be stated. Their proofs are a simple, but lengthy, application of diagram chasing. Lemma 1.7 Given two exact sequences and a set of morphisms be- tween the groups comprising them such that the following diagram commutes, f1- f2- f3- f4- G1 G2 G3 G4 G5 g1 g2 g3 g4 g5 ? ? ? ? ? f5- f6- f7- f8- G6 G7 G8 G9 G10 4 the five lemma states that if g1; g2; g4; and g5 are isomorphisms, then g3 must be one as well. A corollary to this regarding short exact sequences is as follows. Given two short exact sequences morphisms between the groups that compose them such that the following diagram commutes, - f1- f2- - 1 G1 G2 G3 1 g1 g2 g3 ? ? ? - f3- f4- - 1 G4 G5 G6 1 the short five lemma states that if g1 and g3 are isomorphisms, then so is g2. The strong five lemma replaces those isomorphisms with injective maps. Definition 1.8 A fibre bundle is a quadruple (E; X; F; π), where E is called the total space, X is the base space, F is the fibre, and π : E ! X is the projection of E onto X such that there exists an open neigh- borhood U of every point x 2 X which makes the following diagram commute: =∼ π−1(U) / U × F π proj 1 U y where proj1 is the standard projection on the first coordinate. A cov- ering with such open neighborhoods with their respective homeomor- phisms is referred to as a local trivialization of the bundle. 2. Geometric Braids The theory of braids begins with a very intuitive geometrical descrip- tion of the main objects of study. Definition 2.1 A geometric braid on n strings is a subset β ⊂ R2 × [0; 1] such that β is composed of n disjoint topological intervals (maps from the unit interval into a space). Furthermore, β must satisfy the following conditions: 1.) β \ (R2 × f0g) = f(1; 0; 0); (2; 0; 0); :::; (n; 0; 0)g 2.) β \ (R2 × f1g) = f(1; 0; 1); (2; 0; 1); :::; (n; 0; 1)g 3.) β \ (R2 × ftg) consists of n points for all t 2 [0; 1] 5 2 4.) For any string in β, there exists a projection proji : R × [0; 1] ! [0; 1] taking that string homeomorphically to the unit interval. Two braids are considered isotopic if one may be deformed into the other in a manner such that each of the intermediate steps in this deformation yields a geometric braid. Given an arbitrary braid β, tracing along the strands, one finds that the 0 endpoints are permuted relative to the 1 endpoints. This permutation is the called the underlying permutation of β. A braid is called pure if its underlying permutation trivial (i.e., its endpoints are unpermuted). 3. The Braid Group of a General Manifold Definition 3.1 Taking R2 to be the euclidian plane, the classical braid 2 2 group on n strings is simply π1B0;nR and, analogously, π1F0;nR is the corresponding pure braid group. Clearly, the elements of each braid group thus defined are simply geo- metric braids, for one may think of these braids as the graph of a 2 map from [0; 1] into the space B0;nE (starting and ending at the same point in the space). Thus, taking a base of n distinct points in R2, one may represent any element of the classical braid group as a geometric braid. Analogously for the pure braid group, but each string must start and end at the same point. Composition of braids is simply given by stacking one braid atop another. Definition 3.2 Given any manifold M, and a basepoint in its configu- ration space, its braid group is given by π1B0;nM. Its pure (unpermuted) braid group is given by π1F0;nM. Although this fact will not be proven (its proof is quite long and in- volved, and is not the focus of this paper), it is pretty clear that for all dimensions greater than 2, the braid group is trivial. A non-rigorous proof may be given as follows. For any manifold, braids comprise one dimensional objects living in dim M + 1 dimensions.